Hessian comparison theorems via these geometric inequalities lead to
embedding theorems for weighted Sobolev spaces of functions on Riemannian manifolds.
Let \(M\)  be manifold as in Theorem REF , or Theorem REF , or Theorem REF . Following Costa's notation [1]}, or [2]}, we let \(D_{\gamma }^{1,2}(M)\)  denote the completion of \(C_{0}^{\infty }(M\backslash \left\lbrace  x_{0}\right\rbrace )\) 
with respect to the norm
\(\left|\left|u\right|\right|_{D_{\gamma }^{1,2}(M)}:=\left( \underset{M}{\int }\frac{\left|\nabla u\right|^{2}}{r^{2\gamma }}dv\right) ^{\frac{1}{2}}\, .\) 
